6.1: Angle Pairs

Introduction

House building and Tipis

In Mrs. Patterson’s World Cultures class the students have just started studying house building. Mrs. Patterson explained that house building doesn’t just pertain to the kinds of houses that we live in, but to houses around the world both past and present. The students are going to work on a project on a specific type of house.

Jaime is very excited. She has always been interested in Native Americans, so she has chosen to work on a tipi. Jaime selected one of the books that Mrs. Patterson brought in and began to leaf through the pages looking at all of the different types of tipis constructed.

“Tipis?” Mrs. Patterson asked looking over Jaime’s shoulder.

“Yes, I want to design and build one for part of my project,” Jaime explained.

“That’s wonderful. You will need to use a lot of math to accomplish that too,” Mrs. Patterson stated.

Jaime hadn’t thought about the math involved in building a tipi. But as she looked through the pages on designing and building tipis, she noticed that there were a lot of notes on different angles.

One of the types of angles mentioned was a complementary pair of angles. Another was a supplementary pair of angles. These angles were important to figure out when stitching the liner of the tipi together.

Jaime is puzzled. She can’t remember how to identify a complementary or a supplementary pair of angles.

This lesson is all about angle pairs and relationships. Pay close attention to the information in the lesson and at the end you will be able to help Jaime to identify these angle pairs.

What You Will Learn

In this lesson you will learn how to understand the following skills.

Identify angle pairs as complementary, supplementary or neither.

Identify adjacent and vertical angles formed by intersecting lines.

Identify intersecting, parallel or perpendicular lines in a plane.

Find measures of angle pairs using known relationships and sufficient given information.

Teaching Time

I. Identify Angle Pairs as Complementary, Supplementary or Neither

In this lesson, we will begin to examine angles that are formed by different types of lines.

Anangleis measurement of the space created when lines intersect. Here is an example of the angles formed when two lines intersect. You can see that there are four angles created in this drawing and that they are labeled 1 – 4.

We have reviewed some lines and that angles are created when lines intersect. Sometimes, the way that the lines intersect can create anangle pair. This is when two special angles are formed and these angles have a special relationship. Let’s look at some angle pairs.

The two basic forms of angle pairs are called complementary and supplementary angles.

Complementary anglesare two angles whose measurements add up to exactly \begin{align*}90^{\circ}\end{align*}. In other words, when we put them together they make a right angle. Below are some pairs of complementary angles.

Supplementary anglesare two angles whose measurements add up to exactly \begin{align*}180^{\circ}\end{align*}. When we put them together, they form a straight angle.Astraight angleis a line. Take a look at the pairs of supplementary angles below.

Once you know how to identify the angle pairs, you will be able to classify angle pairs as supplementary, complementary or neither. Let’s look at an example.

Example

Classify the following pairs of angles as either complementary or supplementary.

Now let’s look at how we can identify the angle pairs.

First, look at the first pair of angles labeled \begin{align*}a\end{align*}. We can see that the measure of the angles in this pair is 30 and 60 degrees. We know that the sum of complementary angles is \begin{align*}90^{\circ}\end{align*}. Therefore, we can identify this angle pair as complementary angles.

Now look at the second angle pair labeled \begin{align*}b\end{align*}. We can see that the measure of the angles in this pair is 110 and 70. The sum of these two angles is \begin{align*}180^{\circ}\end{align*}. These angles are supplementary angles.

Note: The word “supplementary” or “complementary” refers to the relationship between the two angles.

Sometimes, a pair of angles will be neither complementary nor supplementary. Let’s look at an example.

The sum of these angles is \begin{align*}70^{\circ}\end{align*}. 70 is not 90 nor is it 180, so this angle pair is neither complementary nor supplementary.

Go back and write all of the vocabulary words from this section in your notebooks. Draw a small example of each word next to its definition.

II. Identify Adjacent and Vertical Angles Formed by Intersecting Lines

When lines intersect, they create special relationships between the angles that they form. Once we understand these relationships, we can use them to find the measure of angles formed by the intersecting lines.

Adjacent anglesare angles that share the same vertex and one common side. If they combine to make a straight line, adjacent angles must add up to \begin{align*}180^{\circ}\end{align*}. The word “adjacent” means “next to” that can help you to remember adjacent angles.

Below, angles 1 and 2 are adjacent. Angles 3 and 4 are also adjacent.

Can you see that angles 1 and 2, whatever their measurements are, will add up to \begin{align*}180^{\circ}\end{align*}? This is true for angles 3 and 4, because they also form a line. But that’s not all. Angles 1 and 4 also form a line. So do angles 2 and 3. These are also pairs of adjacent angles. Because the adjacent angle pairs form lines, we can also say that they are supplementary. They must add up to \begin{align*}180^{\circ}\end{align*}.

As you work through this lesson, you will find that some information leads you to other information. Here is the first example of that. Because adjacent angles form a straight line, they are also supplementary. The sum of the angles is \begin{align*}180^{\circ}\end{align*}.

Notice that when there are two angles next to each other, there are also two angles diagonally across from each other. These are called vertical angles.Vertical anglesare angles that are diagonally across from each other and have the same measure.

These relationships always exist whenever any two lines intersect. Look carefully at the figures below. Understanding the four angles formed by intersecting lines is a very important concept in geometry.

Example

Identify the vertical angles and the adjacent angles in the diagram below.

First, think back to the definition of adjacent and vertical angles.

Adjacent angles form a straight line. They are supplementary angles. We can see from the diagram that angles 1 and 3 are adjacent. Angles 2 and 4 are also adjacent.

Vertical angles are diagonal from each other and have the same measure. In this case, angles 1 and 4 are vertical. Angles 2 and 3 are also vertical angles.

You will use this information again when problem solving, but now, let’s look at types of lines.

III. Identify Intersecting, Parallel and Perpendicular Lines in a Plane

In other math classes, you learned about different types of lines. Lines exist in space. Two lines intersect when they cross each other. Because all lines are straight, intersecting lines can only cross each other once.

There are parallel lines, intersecting lines and perpendicular lines. Let’s start by briefly reviewing these terms and then we can look at the angles formed when these lines intersect.

Types of Lines

Parallel Linesare lines that are an equal distance apart. This means that these lines will never intersect.

Intersecting linesare lines that cross at some point.

Perpendicular linesare lines that intersect at a \begin{align*}90^{\circ}\end{align*} angle.

Keep all of this information in mind as we now apply what we have learned to problem solving.

IV. Find Measures of Angle Pairs using Known Relationships and Sufficient Given Information

Now that you have learned some information about angle pairs and their relationships, you can use what you have learned to find the measures of missing angles. Let’s review what we have just learned.

Supplementary angles are two angles that form a straight line, and their sum is always \begin{align*}180^{\circ}\end{align*}. Complementary angles together form a right angle and have a sum of \begin{align*}90^{\circ}\end{align*}.

Adjacent angles are next to each other. When they form a line, their sum is \begin{align*}180^{\circ}\end{align*}.

Vertical angles are directly opposite each other. They are equal.

Here is our first example.

Example

Fill in the figure below with the angle measures for all of the angles shown.

First, notice that we only have one angle to go on. This angle measures 70 degrees. However, that is enough information to figure out all of the other angles in this diagram. We can use the information that we know about angles to figure the measures of these angles out.

Let’s begin with adjacent angles. Angle \begin{align*}b\end{align*} is adjacent to the 70 degree angle. Since we know that adjacent angles form a straight line, the sum of the two angles is \begin{align*}180^{\circ}\end{align*}.

We can write this equation.

\begin{align*}180 = 70 + b\end{align*}

We know that \begin{align*}b\end{align*} is equal to \begin{align*}110^{\circ}\end{align*}.

Next, we can work on the vertical angles. Angle \begin{align*}c\end{align*} is vertical with angle \begin{align*}b\end{align*}. Vertical angles have the same measure, so the measure of angle \begin{align*}c\end{align*} is also \begin{align*}110^{\circ}\end{align*}.

Angle \begin{align*}a\end{align*} is vertical with the \begin{align*}70^{\circ}\end{align*} given angle, we know that this one is also \begin{align*}70^{\circ}\end{align*}.

Using our known information, we have figured out the measures of all of the missing angles.

Now let’s take what we have learned and apply it to the problem from the introduction.

Real-Life Example Completed

House building and Tipis

Here is the original problem once again. Reread it and then write a definition to describe both complementary and supplementary angles. There are two parts to your answer.

In Mrs. Patterson’s World Cultures class the students have just started studying house building. Mrs. Patterson explained that house building doesn’t just pertain to the kinds of houses that we live in, but to houses around the world both past and present. The students are going to work on a project on a specific type of house.

Jaime is very excited. She has always been interested in Native Americans, so she has chosen to work on a tipi. Jaime selected one of the books that Mrs. Patterson brought in and began to leaf through the pages looking at all of the different types of tipis constructed.

“Tipis?” Mrs. Patterson asked looking over Jaime’s shoulder.

“Yes, I want to design and build one for part of my project,” Jaime explained.

“That’s wonderful. You will need to use a lot of math to accomplish that too,” Mrs. Patterson stated.

Jaime hadn’t thought about the math involved in building a tipi. But as she looked through the pages on designing and building tipis, she noticed that there were a lot of notes on different angles.

One of the types of angles mentioned was a complementary pair of angles. Another was a supplementary pair of angles. These angles were important to figure out when stitching the liner of the tipi together.

Jaime is puzzled. She can’t remember how to identify a complementary or a supplementary pair of angles.

Remember there are two parts to your answer.

Solution to Real – Life Example

Jaime needs to understand the difference between complementary and supplementary angle pairs. First, notice that the word “pair” refers to two, so we are talking about two angles.

Here are the definitions.

Complementary Angles – are two angles whose sum is \begin{align*}90^{\circ}\end{align*}.

Supplementary angles – are two angles whose sum is \begin{align*}180^{\circ}\end{align*}.

Complementary angles form a right angle and supplementary angles form a straight line.

Vocabulary

Here are the vocabulary words that are found in this lesson.

Parallel lines

lines that are an equal distance apart and will never intersect.

Intersecting lines

lines that cross at one point.

Perpendicular lines

lines that intersect at a \begin{align*}90^{\circ}\end{align*} angle and form two or more \begin{align*}90^{\circ}\end{align*} angles.

Angle

the measure of the space formed by two intersecting lines.

Straight angle

is a straight line equal to \begin{align*}180^{\circ}\end{align*}.

Angle Pairs

the relationship formed by two angles.

Complementary Angles

two angles whose sum is \begin{align*}90^{\circ}\end{align*}.

Supplementary Angles

two angles whose sum is \begin{align*}180^{\circ}\end{align*}.

Adjacent Angles

angles that are next to each other and whose sum is \begin{align*}180^{\circ}\end{align*}.

Vertical Angles

angles that are diagonally across from each other and whose sum is \begin{align*}90^{\circ}\end{align*}.

Time to Practice

Directions: Write the definitions for the following types of lines.

Parallel lines

Intersecting lines

Perpendicular lines

Directions: Answer the following questions about different types of lines.

What is the symbol for parallel lines?

What is the symbol for perpendicular lines?

An intersection on a highway is an example of what type of lines?

A four way stop is an example of what type of lines?

Is it possible for intersecting lines to also be considered parallel or perpendicular?

Directions: If the following angle pairs are complementary, then what is the measure of the missing angle?

\begin{align*}\angle{A}&=55^{\circ}\\
\angle{B}&= ?\end{align*}

\begin{align*}\angle{C}&=33^{\circ}\\
\angle{D}&= ?\end{align*}

\begin{align*}\angle{E}&=83^{\circ}\\
\angle{F}&= ?\end{align*}

\begin{align*}\angle{G}&=73^{\circ}\\
\angle{H}&= ?\end{align*}

Directions: If the following angle pairs are supplementary, then what is the measure of the missing angle?